YES

Problem:
 g(c(x1)) -> g(f(c(x1)))
 g(f(c(x1))) -> g(f(f(c(x1))))
 g(g(x1)) -> g(f(g(x1)))
 f(f(g(x1))) -> g(f(x1))

Proof:
 DP Processor:
  DPs:
   g#(c(x1)) -> f#(c(x1))
   g#(c(x1)) -> g#(f(c(x1)))
   g#(f(c(x1))) -> f#(f(c(x1)))
   g#(f(c(x1))) -> g#(f(f(c(x1))))
   g#(g(x1)) -> f#(g(x1))
   g#(g(x1)) -> g#(f(g(x1)))
   f#(f(g(x1))) -> f#(x1)
   f#(f(g(x1))) -> g#(f(x1))
  TRS:
   g(c(x1)) -> g(f(c(x1)))
   g(f(c(x1))) -> g(f(f(c(x1))))
   g(g(x1)) -> g(f(g(x1)))
   f(f(g(x1))) -> g(f(x1))
  TDG Processor:
   DPs:
    g#(c(x1)) -> f#(c(x1))
    g#(c(x1)) -> g#(f(c(x1)))
    g#(f(c(x1))) -> f#(f(c(x1)))
    g#(f(c(x1))) -> g#(f(f(c(x1))))
    g#(g(x1)) -> f#(g(x1))
    g#(g(x1)) -> g#(f(g(x1)))
    f#(f(g(x1))) -> f#(x1)
    f#(f(g(x1))) -> g#(f(x1))
   TRS:
    g(c(x1)) -> g(f(c(x1)))
    g(f(c(x1))) -> g(f(f(c(x1))))
    g(g(x1)) -> g(f(g(x1)))
    f(f(g(x1))) -> g(f(x1))
   graph:
    f#(f(g(x1))) -> f#(x1) -> f#(f(g(x1))) -> g#(f(x1))
    f#(f(g(x1))) -> f#(x1) -> f#(f(g(x1))) -> f#(x1)
    f#(f(g(x1))) -> g#(f(x1)) -> g#(g(x1)) -> g#(f(g(x1)))
    f#(f(g(x1))) -> g#(f(x1)) -> g#(g(x1)) -> f#(g(x1))
    f#(f(g(x1))) -> g#(f(x1)) -> g#(f(c(x1))) -> g#(f(f(c(x1))))
    f#(f(g(x1))) -> g#(f(x1)) -> g#(f(c(x1))) -> f#(f(c(x1)))
    f#(f(g(x1))) -> g#(f(x1)) -> g#(c(x1)) -> g#(f(c(x1)))
    f#(f(g(x1))) -> g#(f(x1)) -> g#(c(x1)) -> f#(c(x1))
    g#(f(c(x1))) -> f#(f(c(x1))) -> f#(f(g(x1))) -> g#(f(x1))
    g#(f(c(x1))) -> f#(f(c(x1))) -> f#(f(g(x1))) -> f#(x1)
    g#(f(c(x1))) -> g#(f(f(c(x1)))) -> g#(g(x1)) -> g#(f(g(x1)))
    g#(f(c(x1))) -> g#(f(f(c(x1)))) -> g#(g(x1)) -> f#(g(x1))
    g#(f(c(x1))) -> g#(f(f(c(x1)))) ->
    g#(f(c(x1))) -> g#(f(f(c(x1))))
    g#(f(c(x1))) -> g#(f(f(c(x1)))) -> g#(f(c(x1))) -> f#(f(c(x1)))
    g#(f(c(x1))) -> g#(f(f(c(x1)))) -> g#(c(x1)) -> g#(f(c(x1)))
    g#(f(c(x1))) -> g#(f(f(c(x1)))) -> g#(c(x1)) -> f#(c(x1))
    g#(g(x1)) -> f#(g(x1)) -> f#(f(g(x1))) -> g#(f(x1))
    g#(g(x1)) -> f#(g(x1)) -> f#(f(g(x1))) -> f#(x1)
    g#(g(x1)) -> g#(f(g(x1))) -> g#(g(x1)) -> g#(f(g(x1)))
    g#(g(x1)) -> g#(f(g(x1))) -> g#(g(x1)) -> f#(g(x1))
    g#(g(x1)) -> g#(f(g(x1))) -> g#(f(c(x1))) -> g#(f(f(c(x1))))
    g#(g(x1)) -> g#(f(g(x1))) -> g#(f(c(x1))) -> f#(f(c(x1)))
    g#(g(x1)) -> g#(f(g(x1))) -> g#(c(x1)) -> g#(f(c(x1)))
    g#(g(x1)) -> g#(f(g(x1))) -> g#(c(x1)) -> f#(c(x1))
    g#(c(x1)) -> f#(c(x1)) -> f#(f(g(x1))) -> g#(f(x1))
    g#(c(x1)) -> f#(c(x1)) -> f#(f(g(x1))) -> f#(x1)
    g#(c(x1)) -> g#(f(c(x1))) -> g#(g(x1)) -> g#(f(g(x1)))
    g#(c(x1)) -> g#(f(c(x1))) -> g#(g(x1)) -> f#(g(x1))
    g#(c(x1)) -> g#(f(c(x1))) -> g#(f(c(x1))) -> g#(f(f(c(x1))))
    g#(c(x1)) -> g#(f(c(x1))) -> g#(f(c(x1))) -> f#(f(c(x1)))
    g#(c(x1)) -> g#(f(c(x1))) -> g#(c(x1)) -> g#(f(c(x1)))
    g#(c(x1)) -> g#(f(c(x1))) -> g#(c(x1)) -> f#(c(x1))
   EDG Processor:
    DPs:
     g#(c(x1)) -> f#(c(x1))
     g#(c(x1)) -> g#(f(c(x1)))
     g#(f(c(x1))) -> f#(f(c(x1)))
     g#(f(c(x1))) -> g#(f(f(c(x1))))
     g#(g(x1)) -> f#(g(x1))
     g#(g(x1)) -> g#(f(g(x1)))
     f#(f(g(x1))) -> f#(x1)
     f#(f(g(x1))) -> g#(f(x1))
    TRS:
     g(c(x1)) -> g(f(c(x1)))
     g(f(c(x1))) -> g(f(f(c(x1))))
     g(g(x1)) -> g(f(g(x1)))
     f(f(g(x1))) -> g(f(x1))
    graph:
     f#(f(g(x1))) -> f#(x1) -> f#(f(g(x1))) -> f#(x1)
     f#(f(g(x1))) -> f#(x1) -> f#(f(g(x1))) -> g#(f(x1))
     f#(f(g(x1))) -> g#(f(x1)) -> g#(f(c(x1))) -> f#(f(c(x1)))
     f#(f(g(x1))) -> g#(f(x1)) -> g#(f(c(x1))) -> g#(f(f(c(x1))))
     f#(f(g(x1))) -> g#(f(x1)) -> g#(g(x1)) -> f#(g(x1))
     f#(f(g(x1))) -> g#(f(x1)) -> g#(g(x1)) -> g#(f(g(x1)))
     g#(g(x1)) -> g#(f(g(x1))) -> g#(g(x1)) -> f#(g(x1))
     g#(g(x1)) -> g#(f(g(x1))) -> g#(g(x1)) -> g#(f(g(x1)))
     g#(c(x1)) -> g#(f(c(x1))) -> g#(f(c(x1))) -> f#(f(c(x1)))
     g#(c(x1)) -> g#(f(c(x1))) -> g#(f(c(x1))) -> g#(f(f(c(x1))))
    CDG Processor:
     DPs:
      g#(c(x1)) -> f#(c(x1))
      g#(c(x1)) -> g#(f(c(x1)))
      g#(f(c(x1))) -> f#(f(c(x1)))
      g#(f(c(x1))) -> g#(f(f(c(x1))))
      g#(g(x1)) -> f#(g(x1))
      g#(g(x1)) -> g#(f(g(x1)))
      f#(f(g(x1))) -> f#(x1)
      f#(f(g(x1))) -> g#(f(x1))
     TRS:
      g(c(x1)) -> g(f(c(x1)))
      g(f(c(x1))) -> g(f(f(c(x1))))
      g(g(x1)) -> g(f(g(x1)))
      f(f(g(x1))) -> g(f(x1))
     graph:
      f#(f(g(x1))) -> f#(x1) -> f#(f(g(x1))) -> g#(f(x1))
      f#(f(g(x1))) -> f#(x1) -> f#(f(g(x1))) -> f#(x1)
      f#(f(g(x1))) -> g#(f(x1)) -> g#(g(x1)) -> g#(f(g(x1)))
      f#(f(g(x1))) -> g#(f(x1)) -> g#(g(x1)) -> f#(g(x1))
      g#(c(x1)) -> g#(f(c(x1))) -> g#(f(c(x1))) -> g#(f(f(c(x1))))
      g#(c(x1)) -> g#(f(c(x1))) -> g#(f(c(x1))) -> f#(f(c(x1)))
     SCC Processor:
      #sccs: 1
      #rules: 1
      #arcs: 6/64
      DPs:
       f#(f(g(x1))) -> f#(x1)
      TRS:
       g(c(x1)) -> g(f(c(x1)))
       g(f(c(x1))) -> g(f(f(c(x1))))
       g(g(x1)) -> g(f(g(x1)))
       f(f(g(x1))) -> g(f(x1))
      Arctic Interpretation Processor:
       dimension: 1
       interpretation:
        [f#](x0) = x0,
        
        [f](x0) = x0,
        
        [g](x0) = 1x0,
        
        [c](x0) = 7x0
       orientation:
        f#(f(g(x1))) = 1x1 >= x1 = f#(x1)
        
        g(c(x1)) = 8x1 >= 8x1 = g(f(c(x1)))
        
        g(f(c(x1))) = 8x1 >= 8x1 = g(f(f(c(x1))))
        
        g(g(x1)) = 2x1 >= 2x1 = g(f(g(x1)))
        
        f(f(g(x1))) = 1x1 >= 1x1 = g(f(x1))
       problem:
        DPs:
         
        TRS:
         g(c(x1)) -> g(f(c(x1)))
         g(f(c(x1))) -> g(f(f(c(x1))))
         g(g(x1)) -> g(f(g(x1)))
         f(f(g(x1))) -> g(f(x1))
       Qed